Classification of 3-dimensional integrable scalar discrete equations

TL;DR

This paper classifies all integrable scalar discrete equations on a 3D lattice, showing that under symmetry constraints, the only nontrivial case is the known dBKP-system, thus advancing understanding of integrability in discrete systems.

Contribution

It provides a complete classification of integrable 3D scalar discrete equations with symmetry invariance, identifying the dBKP-system as the unique nontrivial example.

Findings

Only the dBKP-system is nontrivial and integrable under symmetry constraints.

The classification confirms the uniqueness of the dBKP-system in this context.

The work refines previous results with small corrections and clarifications.

Abstract

We classify all integrable 3-dimensional scalar discrete quasilinear equations Q=0 on an elementary cubic cell of the 3-dimensional lattice. An equation Q=0 is called integrable if it may be consistently imposed on all 3-dimensional elementary faces of the 4-dimensional lattice. Under the natural requirement of invariance of the equation under the action of the complete group of symmetries of the cube we prove that the only nontrivial (non-linearizable) integrable equation from this class is the well-known dBKP-system. (Version 2: A small correction in Table 1 (p.7) for n=2 has been made.) (Version 3: A few small corrections: one more reference added, the main statement stated more explicitly.)